The present invention relates generally to the synchronization of nonlinear systems and more particularly to a system which allows the synchronization of one nonlinear dynamical or chaotic system to another nonlinear dynamical or chaotic system insensitive to attenuation in the driving signal.
A synchronized nonlinear system can be used as an information transfer system. The transmitter, responsive to an information signal, produces a drive signal for transmission to the receiver. An error detector compares the drive signal and the output signal produced by the receiver to produce an error signal indicative of the information contained in the information signal.
It is known to those skilled in the art that a nonlinear dynamical system can be driven (the response) with a signal from another nonlinear dynamical system (the drive). With such a configuration the response system actually consist of duplicates of subsystems of the drive system, which are cascaded and the drive signal, or signals, come from parts of the drive system that are included in the response system.
A system with extreme sensitivity to initial conditions is considered chaotic. The same chaotic system started at infinitesimally different initial conditions may reach significantly different states after a period of time. Lyapanov exponents (also known in the art as "characteristic exponents") measure this divergence. A system will have a complete set of Lyapunov exponents, each of which is the average rate of convergence (if negative) or divergence (if positive) of nearby orbits in phase space as expressed in terms of appropriate variables and components.
Sub or Conditional Lyapunov exponents are characteristic exponents which depend on the signal driving the system. It is also known to those skilled in the art that, if the sub-Lyapunov, or conditional Lyapunov, exponents for the driven response system are all negative, then all signals in the response system will converge over time or synchronize with the corresponding signals in the drive. When the response system is driven with the proper signal from the drive system, the output of the response system is identical to the input signal. When driven with any other signal, the output from the response is different from the input signal.
In brief, a dynamical system can be described by the ordinary differential equation EQU .alpha.(t)=f(.alpha.) (1)
The system is then divided into two subsystems. .alpha.=(.beta.,.chi.); EQU .beta.=g(.beta.,.chi.) EQU .chi.=h(.beta.,102 ) (2)
where .beta.=(.alpha..sub.l. . . .alpha..sub.n), g=(f.sub.l (.alpha.) . . . f.sub.n (.alpha.)), h=(f.sub.n+l (.alpha.) . . . f.sub.m (.alpha.)), .chi.=(.alpha..sub.n+l, . . . .alpha..sub.m), where .alpha., .beta. and .chi. are measurable parameters of a system, for example vectors representing a electromagnetic wave.
The division is arbitrary since the reordering of the .alpha..sub.i variables before assigning them to .beta., .chi. g and h is allowed. A first response system is created by duplicating a new subsystem .chi.' identical to the .chi. system, and substituting the set of variables .beta. for the corresponding .beta.' in the function h, and augmenting Eqs. (2) with this new system, giving EQU .beta.=g(.beta.,.chi.) EQU .chi.=h(.beta.,.chi.) (3) EQU .chi.'=h(.beta.,.chi.')
If all the sub-Lyapunov exponents of the .chi.' system (ie. as it is driven) are less than zero, then [.chi.'-.chi.].fwdarw.0 as t .fwdarw.infinity. The variable .beta. is known as the driving signal. One may also reproduce the .beta. subsystem and drive it with the .chi.' variable, giving EQU .beta.=g(.beta.,.chi.) EQU .chi.=h(.beta.,.chi.) EQU .chi.'=h(.beta.,.chi.') (4) EQU .beta."=g(.beta.",.chi.')
The functions h and g may contain some of the same variables. If all the sub-Lypaunov exponents of the .chi.', .beta." subsystem (ie. as it is driven) are less than 0, then .beta.".fwdarw..beta. as t .fwdarw.infinity. The example of the Equ. (4) is referred to as cascaded synchronization. Synchronization is confirmed by comparing the driving signal .beta. with the signal .beta.".
Generally, since the response system is nonlinear, it will only synchronize to a drive signal with the proper amplitude. If the response system is as some remote location with respect to the drive system, the drive signal will probably be subjected to some unknown attenuation. This attenuation can be problematic to system synchronization.
The present invention is a system design featuring subsystems which are nonlinear and possibly chaotic, but will still synchronize when the drive signal is attenuated or amplified by an unknown amount. This invention builds on the design of two previous inventions, the synchronizing of chaotic systems, U.S. Pat. No. 5,245,660, and the cascading of synchronized chaotic systems, U.S. Pat. No. 5,379,346, both herein incorporated by reference. The present invention extends those principles to situations where the drive signal has been attenuated by some unknown amount. Applicants overcome this limitation by providing for a nonlinear response system that is not amplitude dependent and a separate function that is part of the drive system only that is amplitude dependent.